Plug for a friend's book that is forthcoming later this summer, "Topology: A Categorical Approach". [1]
I can't speak to its contents per se, because there isn't a preview yet, but I can speak to the quality of exposition in the lead author's math blog. [2]
I haven't ever dug too much into category theory for its own sake (usually just one-off chapters or appendices that get included in books on other topics), but my understanding is that it unites a lot of mathematical topics. As such, this book might be of more interest to you than, say, a classical point-set topology text, given your desire to uncover connections. That being said, there may be other category-theory-flavored books on other more strictly algebraic topics that would suit your fancy more.